Plancherel-type estimates and sharp spectral multipliers
type: Item
instance of: scholarly article
title: Plancherel-type estimates and sharp spectral multipliers
zbMATH Open document ID: 1029.43006
DOI: 10.1016/S0022-1236(02)00009-5
author: Xuan Thinh Duong
author: E. M. Ouhabaz
author: Adam S. Sikora
published in: Journal of Functional Analysis
publication date: 2003-03-26T00:00:00Z
review text: General spectral multiplier theorems for self-adjoint positive definite operators on \(L^{2}\left( X,\mu \right)\) are investigated considering an arbitrary open subset \(X\) of a space of homogeneous type. The spectral multipliers are studied in a general setting of abstract operators rather than in a specific setting of group invariant operators acting on Lie groups. It is shown that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the \(L^{2}\) norm of the kernel of spectral multipliers and the Gaussian bounds of the corresponding heat kernel. The critical exponent for the Riesz summability is also studied. Several examples including elliptic differential operators on compact manifolds, the Hermite and Laguerre expansions and scattering-type operators on \(R^{3}\) are given in order to motivate the introduction of Plancherel-type estimates.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 1888377
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: Plancherel-type estimates
zbMATH Keywords: spectral multipliers
zbMATH Keywords: scattering-type operators
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
cites work: On the asymptotic behavior of spectral functions and resolvent kernels of elliptic operators
cites work: Spectral Multipliers on Lie Groups of Polynomial Growth
cites work: Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten
cites work: Brownian motion on the Sierpinski gasket
cites work: Gaussian estimates and \(L^p\)-boundedness of Riesz means
cites work: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
cites work: The weak type \(L^ 1\) convergence of eigenfunction expansions for pseudodifferential operators
cites work: Weak type (1,1) bounds for rough operators
cites work: L p Bounds for Spectral Multipliers on Nilpotent Groups
cites work: Riesz transforms for $1\le p\le 2$
cites work: Q4881087
cites work: A spectral multiplier theorem for a sublaplacian on SU\((2)\)
cites work: Q4041031
cites work: Uniformly elliptic operators with measurable coefficients
cites work: \(H^ p\)-multipliers on stratified groups
cites work: The spectrum of positive elliptic operators and periodic bicharacteristics
cites work: Semigroup kernels, Poisson bounds, and holomorphic functional calculus
cites work: From the \(L^ 1\) norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups
cites work: Singular integral operators with non-smooth kernels on irregular domains
cites work: Complex multiplicative perturbations of elliptic operators: heat kernel bounds and holomorphic functional calculus
cites work: Inequalities for strongly singular convolution operators
cites work: Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28
cites work: Functional calculus for infinitesimal generators of holomorphic semigroups
cites work: Almost everywhere summability of eigenfunction expansions associated to elliptic operators
cites work: A multiplier theorem for Schrödinger operators
cites work: Multiplier theorem on generalized Heisenberg groups
cites work: Estimates for translation invariant operators in \(L^p\) spaces
cites work: The spectral function of an elliptic operator
cites work: Q3671564
cites work: Q3894695
cites work: Analysis on fractals
cites work: Mean Cesaro summability of Laguerre expansions and norm estimates with shifted parameter
cites work: Vector-valued multipliers on stratified groups
cites work: Q3338673
cites work: Q5337441
cites work: Mean Convergence of Hermite and Laguerre Series. I
cites work: On spectral multipliers for Heisenberg and related groups
cites work: Gaussian Estimates and Holomorphy of Semigroups
cites work: Q4191141
cites work: Q4001537
cites work: On the boundedness of functions of (pseudo-) differential operators on compact manifolds
cites work: Q4896968
cites work: On the 𝐿²→𝐿^{∞} norms of spectral multipliers of “quasi-homogeneous” operators on homogeneous groups
cites work: Imaginary powers of Laplace operators
cites work: Q3891103
cites work: An estimate for a first-order Riesz operator on the affine group
cites work: On the convergence of Riesz means on compact manifolds
cites work: Q4693724
cites work: Q5607484
cites work: Q3142876
cites work: Twisted convolution and Riesz means
cites work: Weak-type endpoint bounds for Riesz means
cites work: Summability of Hermite Expansions. I
cites work: Q3137751
cites work: Q4002250
full work available at URL: s0022-1236(02)00009-5
OpenAlex ID: W2083947946
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